On the blowup of multidimensional semilinear heat equations
On the boundary ergodic problem for fully nonlinear equations in bounded domains with general nonlinear Neumann boundary conditions
On the Caginalp system with dynamic boundary conditions and singular potentials
This article is devoted to the study of the Caginalp phase field system with dynamic boundary conditions and singular potentials. We first show that, for initial data in , the solutions are strictly separated from the singularities of the potential. This turns out to be our main argument in the proof of the existence and uniqueness of solutions. We then prove the existence of global attractors. In the last part of the article, we adapt well-known results concerning the Łojasiewicz inequality in...
On the Cauchy problem for a class of parabolic equations with variable density
The well-posedness of the Cauchy problem for a class of parabolic equations with variable density is investigated. Necessary and sufficient conditions for existence and uniqueness in the class of bounded solutions are proved. If these conditions fail, sufficient conditions are given to ensure well-posedness in the class of bounded solutions which satisfy suitable constraints at infinity.
On the Convective Cahn-Hilliard Equation
The author studies the convective Cahn-Hilliard equation. Some results on the existence of classical solutions and asymptotic behavior of solutions are established. The instability of the traveling waves is also discussed.
On the differential system governing flows in magnetic field with data in .
On the eigenfunction expansion method for semilinear dissipative equations in bounded domains and the Kuramoto-Sivashinsky equation in a ball
Presented herein is a method of constructing solutions of semilinear dissipative evolution equations in bounded domains. For small initial data this approach permits one to represent the solution in the form of an eigenfunction expansion series and to calculate the higher-order long-time asymptotics. It is applied to the spatially 3D Kuramoto-Sivashinsky equation in the unit ball B in the linearly stable case. A global-in-time mild solution is constructed in the space , s < 2, and the uniqueness...
On the existence of solutions of nonlinear infinite systems of parabolic differential-functional equations.
On the expansion of starshaped hypersurfaces by symmetric functions of their principal curvatures.
On the finite dimension of attractors of doubly nonlinear parabolic systems with l-trajectories
This paper is concerned with the asymptotic behaviour of a class of doubly nonlinear parabolic systems. In particular, we prove the existence of the global attractor which has, in one and two space dimensions, finite fractal dimension.
On the Fourier's first problem for a system of non-linear parabolic equations with unbounded coefficients
On the Hölder Continuity of Bounded Weak Solutions of Quasilinear Parabolic Systems.
On the Hölder continuity of solutions of nonlinear parabolic systems
Non-linear second order parabolic systems in the divergent form are considered. It is proved that under some restrictions on the modulus of ellipticity, all weak solutions are continuous.
On the Hölder continuity of weak solutions to nonlinear parabolic systems in two space dimensions
We prove the interior Hölder continuity of weak solutions to parabolic systems (), where the coefficients are measurable in , Hölder continuous in and Lipschitz continuous in and .
On the interior differentiability of weak solutions of parabolic systems with quadratic growth nonlinearities
On the jump set of solutions of the total variation flow
On the Kuramoto-Sivashinsky equation in a disk
We consider the first initial-boundary value problem for the 2-D Kuramoto-Sivashinsky equation in a unit disk with homogeneous boundary conditions, periodicity conditions in the angle, and small initial data. Apart from proving the existence and uniqueness of a global in time solution, we construct it in the form of a series in a small parameter present in the initial conditions. In the stable case we also obtain the uniform in space long-time asymptotic expansion of the constructed solution and...
On the local behaviour of solutions of a certain class of doubly nonlinear parabolic equations.
On the local behaviour of solutions of degenerate parabolic equations with measurable coefficients
On the local integrability and boundedness of solutions to quasilinear parabolic systems.